Trying to understand some basic representation theory I came across the following saying.
"Since the representation of $sl(2;C)$ has to be finite dimensional, there must exist an integer $n \in N_0$ with $(J_+)^{n+1}|u> = 0$, and $(J_+)^n|u> \ne 0$."
Key for this argument is that the representation is finite dimensional. However, it is not obvious to me what it has to be, and I would greatly appreciate any explanation.
There are well-known infinite-dimensional modules for the Lie algebra $L=\mathfrak{sl}_2(\Bbb C)$. We start with an action of $L$ on the polynomial ring $\Bbb C[X,Y]$ and pass to the subspace of homogeneous polynomials.
References:
Infinite dimensional $sl(2,\mathbb{C})$-modules
Proof check - infinite-dimensional $\mathfrak{sl}(2, \mathbb{F})$-module